By a variety, we mean a one-sorted finitary variety of universal algebras. By a regular category, we mean a regular category in the sense of [1]}, i.e., a finitely complete category with coequalizers of kernel pairs and pullback stable regular epimorphisms. Regular categories have been introduced as a context where finite limits and regular epimorphisms behave in a similar way as finite limits and surjections behave in the category of sets. In particular, every variety is a regular category. By a pointed category, we mean a category with a zero object, i.e., an object which is both terminal and initial. A variety is pointed if and only if its algebraic theory contains a unique constant term.
